$\sum\limits_{n=1}^{\infty}\dfrac{\cos(n)+2}{n}$ Does the integral test apply to the series? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: $\dfrac{\cos(x)+2}{x}$ is continuous and positive for all $x\geq 1$. To find whether it's always decreasing for $x\geq1$, let's consider its derivative. $\dfrac{d}{dx}\left( \dfrac{\cos(x)+2}{x} \right)=-\dfrac{x\sin(x) + \cos(x) + 2}{x^2}$ Some values of $x$, such as $x=2k\pi +\dfrac{3\pi}{2}$ for any nonnegative integer $k$, will make $x \sin x <-3$, which will make the derivative positive. So $\dfrac{d}{dx}\left( \dfrac{\cos(x)+2}{x} \right)$ is positive for some $x>1$, which means $\dfrac{\cos(x)+2}{x}$ is increasing at those points. In conclusion, the integral test does not apply to the series.